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A critical point for random graphs with a given degree sequence. (English) Zbl 0823.05050
Let $$\lambda_ 0,\lambda_ 1,\dots$$ be a sequence of non-negative real numbers summing to 1. This paper considers random graphs with $$n$$ vertices and approximately $$\lambda_ i n$$ vertices of degree $$i$$. Let $$Q$$ be the sum of the quantities $$i(i- 2)\lambda_ i$$ over $$i= 1,2,\dots$$. It is shown (under some restrictions) that if $$Q> 0$$ then with probability $$\to 1$$ as $$n\to \infty$$, such graphs have a ‘giant component’, whereas if $$Q< 0$$ then all components are small. This result and the proof method yield new insight on the classical ‘double-jump’ threshold for usual random graphs, and can be applied in investigations concerning the chromatic number of sparse random graphs.

##### MSC:
 05C80 Random graphs (graph-theoretic aspects) 60C05 Combinatorial probability
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